A geometric program:

\begin{align} \min_{x}\quad & f_{0}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x) \leq 1, i= 1\dots m \\ & h_{i}\qty(x) = 1, i = 1 \dots p \end{align}

where \(f_{i}\) is posynomial, and \(h_{i}\) monomial. Notice that taking a log of this thing transforms the monomial into an affine function in \(\log \qty(x)\), and into a logsumexp for posynomials. This also implies that solving for optimal \(\log \qty(x)\), which is same as solving for \(x\) for positive \(x\), is convex problem.

monomial function

\begin{equation} f\qty(x) = cx_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{n}^{a_{n}} \end{equation}

with \(c > 0\), exponent \(a_{i}\) is any real. This term is not used in this way when not in the context of geometric programming.

\(\text{dom}\qty(f) = \mathbb{R}_{++}^{n}\).

posynomial function

\begin{equation} f\qty(x) = \sum_{k=1}^{k} c_{k}x_{1}^{a_{1k}} x_{2}^{a_{2k}} \dots x_{n}^{a_{nk}} \end{equation}

\(\text{dom}\qty(f) = \mathbb{R}_{++}^{n}\).